For positive integers n, if 4 aₙ=(n²+5 n+6) and Sₙ=Σₖ₌₁ⁿ(1/aₖ), then the value of 507 S₂₀₂₅ is :

Solution

Maths sir · 4 steps

Factor n²+5n+6 into (n+2)(n+3), and the sum telescopes almost to nothing.

1. Factor the term. n²+5n+6 = (n+2)(n+3), so aₖ = (k+2)(k+3)/4 and 1/aₖ = 4/(k+2)(k+3).
2. Split into differences. The two factors differ by 1, so 1/(k+2)(k+3) = 1/(k+2) - 1/(k+3). Add these from k=1 to n, and each -1/(k+3) cancels the next +1/(k+3). Only the first and last terms survive.

   Sₙ = 4(1/3 - 1/(n+3))

3. Substitute. Put n = 2025: S₂₀₂₅ = 4(1/3 - 1/2028). Since 2028 = 3 × 676, this is 4 × (676-1)/2028 = 2700/2028 = 675/507.
4. So. 507 S₂₀₂₅ = 507 × 675/507 = 675, which is option (b).

The idea to keep. Whenever a term is 1/(k+a)(k+a+1), write it as 1/(k+a) - 1/(k+a+1). The sum collapses to the first term minus the last.

Answer from the NTA final answer key. Solution by PrepWiser.